I’ve been writing this blog for around 10 months now, and in that time I have covered a lot of different topics in the fields of gaming and game design. We have studied ancient games from history, mobile apps, and even modern console masterpieces, but one type of game I have yet to talk about is the Game Show. Well that changes today.
Deal or No Deal hasn’t been around as long as many other game shows such as The Price is Right or Wheel of Fortune, but it is already one of the most recognizable game show properties of all time. While this show stopped filming in the United States in 2009, it’s presence can still be felt all around the world. There have been over 80 different iterations of this show in countries all around the world, many of which are still on the air. Beyond the shows themselves, Deal or No Deal has also been adapted into numerous video games, as well as a board game edition.
Today, I am going to be examining this game and trying to figure out the optimal strategy in order to get the best deal and win the most money. While there are many versions of this game, I am going to be focusing on the version aired in the United States of America from 2005-2009, and hosted by Howie Mandel. If you are from another part of the world the specifics will likely be different, but hopefully some of the lessons will still apply.
What’s the Deal?
For those who are unfamiliar, the premise behind Deal or No Deal is quite simple. There are 26 briefcases, each of which contain some amount of money, from 1 cent to 1 million dollars. All of the different money amounts are shown on a large board behind the contestants. After choosing a briefcase, the contestant then has to open some of the other briefcases. As the briefcases are opened, the amount of money inside is revealed, and that value is removed from the board. At the end of each round, the “banker” gives the contestant an offer to buy their briefcase, and the contestant gets to choose whether to accept the offer (Deal) or reject it and play another round (No Deal).
This game clearly has a large amount of randomness to it, and may at first appear to be nothing more than blind guessing. However, the player is given enough information to make the decision about whether to take the deal or not. By examining the behavior of the banker, and studying the numbers we can come up with an optimal strategy for getting the best deal out of the game.
Firstly, let’s look at the behavior of the “banker”. The general idea behind the banker is that they want to buy the player’s briefcase for less than it is worth. Because of this, the banker will first calculate the average value of the remaining cases, then offer the player an offer that is generally lower than that average value. At first the banker will usually make an offer that is more significantly below the average value, to encourage the player to refuse the game and keep it going. At the end of the game, on the other hand, the banker might actually make an offer to the player that is above the average, to encourage them to take the deal.
Based on this behavior, it might seem like the obvious strategy is to try and outdo the banker by accepting a deal that is higher than the average of the remaining values. In fact, this is the very strategy that was put forth in this article by Pearsonified. Unfortunately, the actual strategy is not quite that simple. The main reason for this is that the banker doesn’t tend to start making those sorts of offers until late in the game, and by that point you could have lost a lot of value. A much better strategy is to look at each offer in terms of whether you can reasonably expect the offer to go up or down in the next round. If you can expect the offer to go down next round, you should probably take the offer. If it looks like it will go up, then you should play another round.
The idea behind this is simple, but actually figuring out the math can be much more complicated. In order to simplify things, let’s start at the very beginning of the game, with a board full of numbers. The amount of briefcases that you have to open each round starts at 6, and goes down each round. This means that in the first round you have to open 6 briefcases. These 6 are in addition to the one that you selected at the beginning of the game but did not open, which means that you are removing a total of 7 of the 26 options. However, since you don’t open one of the briefcases it is not taken off the board.
This means that at the end of this round there will still be 20 numbers lit up on the board. It is at this point that we might like to know whether we picked “well” or not – basically, did the selections that we made cause the average value of the board to go up or down? In order to examine this, we need a few things. Firstly, we need to know that the average value was before making our picks, and what the average value is now that we have finished choosing our cases.
The first part is pretty easy – there are 26 values, so all we need to do is add them together, and divide by 26. This results in an average value of $131,477.53. Not too shabby! The second part, however, is a little more complicated. While it would be easy to calculate the average value of any particular case, what we are actually concerned with is the average case – that is, can you expect, more often than not, for your new average to be higher than $131,477.53?
In order to answer this question, we would ideally want to generate every possible configuration of numbers that could be remaining after the first round, sum them together, and find the average of all of these cases. However, the number of possible cases is 26 choose 6, which ends up being 230,230 possibilities. Instead of calculating all of these possibilities, lets instead look at the worst and best-case scenarios. Then, I will create a program to approximate the average case.
The best case scenario (at least for this particular situation) is that the player eliminates all of the lowest possible values. Doing so would leave the player with an expected value of $170, 916.25. However, the banker would not offer the full amount, but instead offer some percentage of this amount instead. While the banker’s exact formula is not publicly known, it is known that the percentage goes up over time. Because of this, it is likely that the first round would only be a small fraction of the true value.
The worst case scenario is that all of the high values on the board are eliminated. This would result in a new average of only 13,420.80. This is a much larger shift from the previous value, and the reason for this is that the larger values on the board just make up such a large percentage of the total value.
Finding the average case, however, is much more complicated than finding the best and worst-case scenarios. Instead of calculating all 200,000+ possible combinations and finding the average, I am going to instead perform a Monte Carlo simulation. Basically, I am going to simulate a random round of Deal or No Deal, say, 100,000 times, and use the average of these results. While this figure may not be entirely accurate, it should give us a number that is very close to the actual value. It will also help us in future rounds where the number of combinations explodes even further.
Crunching the numbers for the first several rounds gives the following results
- Round 1 – around $131,500
- Round 2 – also around $131,500
- Round 3 – somewhere around $131,500 …..
I think you can see where this is going, and I probably should have realized it up front – the average of all the subsets is, in the long term, going to converge to the average of the whole set. So does this mean that we are stuck? Well, not entirely. What this means is that, in the long term, you can expect your values to hover around this point. If you have a particularly good first round and eliminate a lot of lower numbers, this makes it more likely for you to hit some of the higher numbers in the next round and bring your offer back down, and vice-versa.
The good news is we can still use this information to our advantage. We know that the dealer’s algorithms increases the percentage of the bid over time, and because the expected value is supposed to even itself out over time, this generally incentivizes playing further rounds. However, this is not always the case, and you will have to tailor your strategy to the individual situation.
One of the biggest factors to keep in mind is just how big a difference some of the larger numbers can make to your average. Hitting lower numbers won’t tend to increase your average all that much (especially early in the game), but hitting higher numbers can result in a severe drop in your total value, as you can see from the best and worst case first round scenarios. Taking all of this into account, I would like to suggest the following strategy:
The overall goal is to beat the banker, which means getting (and accepting) an offer of over $131,500. If you ever receive this offer, I would recommend taking the deal. However, even with great luck it is unlikely for such an offer to be made in the first couple rounds of the game. The first round is completely random, and should be considered the baseline. There is very little reason to ever take the deal in the first round – even if you have very lucky guesses, the offer will likely be such a low percentage of the total that it is not worth it.
From the second round onward, pay close attention to your higher valued numbers – you need numbers over $100,000 dollars in order to have a chance at getting the offer that you are looking for. There are only 6 such numbers on the board, and for each one you should pay attention to how many low values you have remaining on the board. With the $1,000,000 you can have up to 6 low values on the board and still reach the average you want, the $750,000 can have up to 4, the $500,000 can have up to 2 (as can the $400,000), and the $300,000 can handle 1. If you only have the $200,000 remaining, in order to get the average that you are looking for you also need to have at least the $75,000 also on the board.
These numbers are cumulative – suppose you have the $750,000 and the $300,000 on the board. This means that you can have up to 5 lower values and still achieve an offer of over $131,000. However, if you have too many low numbers and not enough high numbers this becomes impossible. Therefore if your ratio becomes too low it is a good idea to accept the offer – otherwise, you risk seeing a severe drop in value.
Until Next Week
That is all I have for this week! I hope you enjoyed this article about Deal or No Deal! If you did, check out the rest of the blog and subscribe on Facebook, Twitter, or here on WordPress so you will always know when I post a new article. If you didn’t, let me know what I can do better in the comments down below. This was a bit different from the usual content that I make, so I would be very interested in hearing all of your thoughts. See you next week!